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Theorem bnj982 31148
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj982.1 (𝜑 → ∀𝑥𝜑)
bnj982.2 (𝜓 → ∀𝑥𝜓)
bnj982.3 (𝜒 → ∀𝑥𝜒)
bnj982.4 (𝜃 → ∀𝑥𝜃)
Assertion
Ref Expression
bnj982 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))

Proof of Theorem bnj982
StepHypRef Expression
1 df-bnj17 31054 . 2 ((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))
2 bnj982.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3 bnj982.2 . . . 4 (𝜓 → ∀𝑥𝜓)
4 bnj982.3 . . . 4 (𝜒 → ∀𝑥𝜒)
52, 3, 4hb3an 2268 . . 3 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
6 bnj982.4 . . 3 (𝜃 → ∀𝑥𝜃)
75, 6hban 2267 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) → ∀𝑥((𝜑𝜓𝜒) ∧ 𝜃))
81, 7hbxfrbi 1893 1 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072  ∀wal 1622   ∧ w-bnj17 31053 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-12 2188 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-bnj17 31054 This theorem is referenced by:  bnj1096  31152  bnj1311  31391  bnj1445  31411
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